Plateau borders cross-section

Fig. 1.9. Scheme explaining the calculation of Plateau border cross-section area. The double hatching in... 

On the other hand a comparison of the ratio between the volumes of real Plateau border and cylindrical border model, obtained by Pertsov et al. [85], with the ratio (rh /rn)2, evaluated from the dodecahedral model [83] indicates that these ratios correlate well only when n > 640 but when n < 300 they differ by almost 30%. However, it should be noted that the method for calculating border volume in [85,86] at low foam expansion ratio differs from the real volume by 10%. Besides, all values of expansion ratio given in the tables of [86] are twice higher. This means that the interpolation equations for calculating the corrections of the volume and hydro- and electrical conductivities in the cylindrical border model must be corrected. The Plateau border cross-section area is determined by Eq. (1.19) when the contact angle between film and border surface is small 6 (or 1 in radians). 

The flow of the continuous-phase liquid out of a foam takes place through the Plateau border channels. The flow of liquid in a single Plateau border channel must therefore be understood before an attempt to model drainage in a foam is made. An expression for the average velocity ( ) of the liquid in a vertical Plateau border channel has been derived for a triangular Plateau border cross-section and is given by [60] ... 

Regular dodecahedral structure of a bubble and cross-sectional view of a plateau border. 

From the above equation, the variation of equilibrium disjoining pressure and the radius of curvature of plateau border with position for a concentrated emulsion can be obtained. If the polarizabilities of the oil, water and the adsorbed protein layer (the effective Hamaker constants), the net charge of protein molecule, ionic strength, protein-solvent interaction and the thickness of the adsorbed protein layer are known, the disjoining pressure II(x/7) can be related to the film thickness using equations 9 -20. The variation of equilitnium film thickness with position in the emulsion can then be calculated. From the knowledge of r and Xp, the variation of cross sectional area of plateau border Qp and the continuous phase liquid holdup e with position can then be calculated using equations 7 and 21 respectively. The results of such calculations for different parameters are presented in the next session. 

Fig. 1.7. Cross-section of Plateau border the definition of radius R is given in Section 4.3.

If the condition for polyhedricity R/r 1 is not fulfilled, the radius of curvature and the area of the cross-section become dependent on the co-ordinates along the length of the Plateau border. Analytical dependence of the radius of curvature on the co-ordinates (the border profile) at different foam expansion ratio is not found. 

Fig. 5. Cross section of plateau border and films in foam.

At the junctions between three films, there are some thicker areas (called Plateau borders) whose cross-section is constant along the edge and has the shape of a curvilinear Plateau triangle formed by arcs of three adjacent circles. 

The most important geometric parameters of polyhedral foam are the length of a Plateau border and its cross-section area. These quantities can be found in [214, 244]. For example, for the pentagonal dodecahedron model, the length of the Plateau border is... 

For monodisperse foam, the area 5b of the cross-section of the Plateau border is... 

An important structural element in a foam is the Plateau border, i.e., the channel having three cylindrical surfaces that is formed between any three adjacent bubbles. Similar channels are formed where two bubbles meet the wall of the vessel containing the foam. Figure 11.3a shows a cross section, and it follows that the Laplace pressure inside the Plateau border is smaller than that in the adjacent films. This means that liquid is sucked from the films to the Plateau borders, whence it can flow away (drain), because the Plateau borders form a connected network. The curvature in the Plateau border is determined by a balance of forces, Laplace pressure versus... 

FIGURE 11.3 Cross sections through Plateau borders and films, (a) Illustration of the difference of the Laplace pressure between border and film, (b) Plateau border at a greater height in the foam. 

In Figure 11.3, two cross sections through a Plateau border with adjacent films are depicted. Assume now that these pictures represent two different situations but at the same height in the foam. Which of the parameters listed below can be the cause of the difference Reason for each parameter whether it would be higher in (a) or in (b) or has no effect ... 

Figure 34-4. Plateau borders in cross section, (a) two capillaries with a film between them, (b) magnified view of one capillary showing the six fold symmetry. (Courtesy - Lemlich, R.. Principles of Foam Fractionation, Wiley-Interscience, New York, NY, 1968)...

In foams with high foam numbers the surface in Gibbs- Plateau borders is close to cylindrical, i.e. has a constant triangular-shaped cross-section with concave sides. The pressure in such cross-section is lowered in comparison with the pressure in foam cells by the amount of o/rcurv, where rcurv is the curvature radius of the border surface (i.e., of the side of triangle). 

Figure 14.16 Plateau borders formed by latex particles as they deform prior to coming into contact, (a) A cross-section through a layer of nearly dry film showing deformed particles, (b) An expanded view of a plateau border with bilayers, showing the radius of curvature and the bilayer thickness, two parameters obtained from the wide-ai neutron-scattering e]q)eriments of ref. 31. (Reprinted with permission from ref. [31]. Copyright 1992 American Chemical Society.)...

Figure 1 (a) Four linear Plateau borders meeting in a tetrahedral Plateau border (b) cross-section through a linear Plateau border and its three associated films and drops. 

FIGURE 6.7 Cross section through contacting foam lamellae. The pressure in the curved Plateau border is reduced. 

The PB channels, which contain most of the liquid, form a complex interconnected network within the framework of Plateau s laws. Fig. 1 shows a typical foam column, a polyhedral bubble, and the cross-section of a typical Plateau border channel formed where three films meet. 

Fig. 2 shows a cross-section of a draining film. The surface is curved at the edges where neighboring films come together to form a Plateau border channel. Due to this curvature, the pressure is smaller at the edges than at the center of the film, and a radial flow is induced, leading to a reduction of the film thickness with time. 

In Eq. (36), 2 is the vertical space coordinate (see Fig. 7) that increases in the downward direction and g, p, p, and p refer to the gravitational acceleration, density, viscosity, and pressure in the continuous phase, respectively. Op is the cross-sectional area of a Plateau border channel and can be computed from... 

The foam/liquid interface moves up (z2 decreases) as liquid drains out of the foam at the bottom. This upward movement occurs because the volume of the foam decreases as liquid leaves the foam. Thus, the rate at which this interface moves up is equal to the rate at which the height of the liquid pool at the bottom increases, which in turn depends on the flow rate of liquid through the Plateau border channels at the bottom. Therefore, if is the cross-sectional area of the foam ... 

Empirical parameter in the Frumkin adsorption equation Average cross-sectional area of a Plateau border channel Foam cross-sectional area Functions used in the calculation of the electrical double-layer force... 

Volumetric flow rate through a Plateau border channel inclined at an angle 9 to the vertical Volumetric flow of feed per unit cross-sectional area at the top of the foam... 

Plateau borders form a hexagonal cross section at plane of observation... 

FIGURE 1.14 Plateau border segment subject to increase in cross-sectional area and decrease in capillary pressure along length Ay and where and Apg are radius of curvature and cross-sectional area of Plateau border, respectively. 

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